ISSN 1303-0485 • eISSN 2148-7561

DOI 10.12738/estp.2015.5.2678 Received | September 30, 2014

Copyright © 2015 EDAM • http://www.estp.com.tr Accepted | September 12, 2015
Educational Sciences: Theory & Practice • 2015 October • 15(5) • 1403-1416 OnlineFirst | October 13, 2015

Investigation of Problem-Solving and Problem-Posing

Abilities of Seventh-Grade Students*
a
Elif Esra Arıkan
Istanbul Girls Anatolian Imam Hatip High School

b
Hasan Ünal
Yıldız Technical University

Abstract
This study aims to examine the effect of multiple problem-solving skills on the problem-posing abilities of gifted
and non-gifted students and to assess whether the possession of such skills can predict giftedness or affect
problem-posing abilities. Participants’ metaphorical images of problem posing were also explored. Participants
were 20 gifted and 85 non-gifted seventh graders, and quantitative and qualitative research methods were
used for data collection and analysis. The relationship between multiple problem-solving skills and giftedness
was investigated, and a strong corre lation between problem solving in multiple ways and problem-posing
abilities was observed in both the gifted and non-gifted students. Moreover, problem solving in multiple ways
was observed in both the gifted and non-gifted students. Metaphorical images were based on the participants’
experiences with problem posing, and they associated their positive or negative metaphors depending on their
problem-posing performance.

Keywords: Problem solving in multiple ways • Problem posing • Seventh-grade students • Metaphor

* This paper was written from the first author’s doctoral dissertation entitled “An Investigation of Mathematical
Problem Solving and Problem Posing Abilities the Students Studying at the Secondary School and
Determining Their Thoughts Concerning Problem Posing by Means of Using Metaphors” that was completed
under the advice of Associate Professor Hasan Ünal at Yıldız Technical University.

a Corresponding author
Elif Esra Arıkan (PhD), Istanbul Girls Anatolian Imam Hatip High School, Şehremini-Fatih, Istanbul Turkey
Research areas: Problem solving; Problem posing; Metaphors; Thinking skills
Email: arikanee@gmail.com

b Assoc. Prof. Hasan Ünal (PhD), Department of Educational Science, Faculty of Education, Yıldız Technical
University, Istanbul 34220 Turkey
Email: hunal@yildiz.edu.tr
Educational Sciences: Theory & Practice

Problem solving and problem posing are accepted In recent years, problem solving and problem
essential components of mathematics education posing have been used as tools for identifying
worldwide. Many studies related to the two concepts students’ thinking and understanding in
have been conducted (Brown & Walter, 1999; mathematics learning. Because problem solving is
Cankoy & Darbaz, 2010; Dede & Yaman, 2005a, a daily necessity, it is a talent that must continually
2005b; Leung, 1996; Silver & Cai, 1996; Schoenfeld, improve to support our continued existence (Skemp,
1992), and research in this area continues to 1987). Because mathematics is not solely a batch of
expand rapidly because of its importance to world numbers, it has to be presented as an approach to
governments. While problem solving is defined memorizing concepts and thus allowing students
as the heart of mathematics education (Cockcraft, to be the ones who research solutions, discover
1982; Dede & Yaman, 2005a, 2005b), problem posing connections and relationships, and realize required
can be identified as one of the coronary vessels. abstractions (Schoenfeld, 1992). Students who
The National Council of Teachers of Mathematics solve problems that they created gain required
(NCTM) (1980) emphasized that students should experience and achieve victory in their discoveries
solve mathematics problems in different ways and (Polya, 1957). Problem solving is a learning process
generate their own problems in given situations. that we go through both at school and throughout
our daily lives (Jonassen, 1997). Students who
If a problem is considered as difficult (Kilpatrick,
follow previously memorized paths in a traditional
1987), problem solving refers to overcoming the
approach do not have the opportunity to create
difficulty. Problem solving is accepted as a central
their own approaches (Hines, 2008). Individuals
activity in education programs in many countries
attempt to solve their problems based on their
such as Australia, Japan, Korea, Singapore, and China,
experience and knowledge even when they do
which were top performers in PISA 2012 (OECD,
not explicitly know the solution. The effort that
2013). Most commonly known problem-solving steps
individuals put into a task is called problem solving
were introduced by Polya (1957), who cited four steps
(Toluk & Olkun, 2001).
to solve a mathematical problem: understanding the
problem (can you state the problem in your own
words?); devising a plan (look for a pattern or equation
or examine related problems); executing the plan
(implementing a strategy and checking operations
and links); and looking back (checking the results).
Many researchers (e.g., Abu-Elwan, 2002) include
problem posing or creating a new problem after final
steps. Furthermore, problem posing is incorporated as
a feature of mathematics teaching in many countries
(e.g., Japan) that employ it as a means of analyzing
problems and enhancing students’ problem-solving
competence (Silver, 1994).
Problem posing in education was introduced by
Freire (1970) for the first time as an alternative to
banking education. Problem posing entails the
generation of a new problem or reformulation of a Figure 1: Interplay among Problem Solving and Mathematics
problem from given situations or problems (English, Creativity, Multiple Problem Solving, Relational vs. Instrumental
Understanding, and Problem Posing.
1997; Grundmeier, 2003; Silver, 1997; Stickles, 2006).
Problem-posing strategies have been examined by The literature reports that problem solving centers
some researchers (Brown & Walter, 1990; Stoyanova around four main constructs:
& Ellerton, 1996). The Turkish Primary Education
1- Mathematical creativity (Mayer, 1970)
Mathematics Curriculum states that “students are able
to solve and pose problems which require calculating 2- Multiple problem solving (Silver, Ghousseini,
with fractions.” Some students cannot comprehend Gosen, Charalambous, & Strawhun, 2005)
fractions because of imperceptions. Therefore, 3- Relational vs. instrumental understanding
students should acquire deep understanding of (Forrester & Chinnappan, 2010)
fractions and feel flexible in studying fractions (Milli
Eğitim Bakanlığı [MEB], 2009). 4- Problem posing (Cai, 1998)

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 Arıkan, Ünal / Investigation of Problem-Solving and Problem-Posing Abilities of Seventh-Grade Students

The four main constructs and their relationship with education, students will have the opportunity to
problem solving are explained in the sections below. conduct their learning activities in flexible, original,
and interrogatory environments (Hinchliffe, 2001;
MEB, 2005, Nelson, 1999). Students’ critical
Problem Posing and Problem Solving capacities can be determined by considering their
Problem posing is a problem-solving activity. It can be capabilities concerning assessing, analyzing, and
defined as the creation of new problems from given creating relationships (Jonassen, 1997). Students
events and situations. For primary school students, with critical thinking skills can comment naturally
problem posing is the center of education in Singapore, on subjects from various aspects and make flexible
which was the highest-performing country in PISA assessments (Vander & Pintrich, 2003). In addition,
2012, and has been adopted as an unchangeable they must organize their knowledge successfully
element in mathematics education because of a and compare and abstract operations accordingly
reform that was enacted in Turkey in 2005 (Ministry (Chance, 1986; Slattery, 1990). The primary
of Education Singapore, 2006, p. 5). Related situations challenge in improving mathematics education
can be, but are not limited to, free, semi-structured, is to improve problem solving by constructing
and structured problem-posing situations that are to knowledge, and it is considered that the notion of
create new problems (Abu-Elwan, 2002; Akay, Soybaş, mathematics as solely classroom based should be
& Argün, 2006; Cankoy & Darbaz, 2010; Lowrie, 2002; abandoned and that it should be emphasized that
Schoenfeld, 1992; Silver & Cai, 1996; Yaman & Dede, mathematics is a tool for real life (MEB, 2009). It
2005). Yaman and Dede (2005) discuss the importance can be taught that problem solving is the center
of problem posing and problem solving in science and of mathematics education programs. Problems
mathematics education. They surveyed the literature are solved using questions that can be answered
and concluded that problem posing and problem in routine, memorable, and formulated ways, i.e.,
solving were strongly related. According to Cankoy students have been known to solve problems using
and Darbaz (2010), people who cannot understand unconventional methods and have also solved
a problem will not be able to find and use suitable open-ended situations (NCTM, 1980).
strategies; moreover, they will not be able to explain Problem posing is a creative activity, and many
what they are doing and why (relational understanding) instruments that measure creativity include
and will ultimately lose the motivation to solve the the problem-finding dimension (Silver, 1994).
problem. The process of problem posing positively Wertheimer (1945) emphasized Einstein’s opinion
affects problem-solving capability (Grundmeier, 2003). that thinking during problem designing and
Thus, similar to problem solving, problem posing problem solving and finding the right question
has also been seen by researchers as the center of are much more important than finding the right
mathematics (Silver, 1997). answer. Jay and Perkins (1997) stated that the key
Conceptual knowledge and operational knowledge to creativity is to produce a new problem or make
are involved in problem solving (Bernardo, 1999). significant modifications to the current problem
It is reported that students who have difficulty to create a new problem. Silver (1997) reported a
solving arithmetic operations also have difficulty similar detection. Although it was not empirically
constructing and solving problems. Factors that proven, Yuan (2009), in his doctoral thesis,
affect problem-solving capability are attitude described how problem posing was used in some
toward the problem, understanding, reasoning, and works to measure creativity, i.e., he studied the
experience (Van de Walle, 1994). Problem solving relationship between problem posing and creativity.
is a sequential activity in traditional education; Originality requires superior talent, and fluency and
students follow what their teachers do (Evancho, flexibility are parts of a natural structure (Leikin,
2000). Mathematics education curricula should be 2009). Solving problems using different approaches
formed solely around problem-solving activities from those of others is a hallmark of creativity.
and determined that individuals must be open- Students must be provided with the opportunity
minded, curious, and patient to improve their to create problems on subjects they are studying,
problem-solving capabilities (NCTM, 1980). From and they should be prompted on those subjects.
this viewpoint, the NCTM places more importance Thus, it will be much clearer how students
on problem solving and mathematical curiosity. attribute meaning to subjects they study (Hiebert
Since 2005, Turkey’s education program, including & Wearne, 2003). Problem posing is maintained as
the structuring approach, has been in progress and an important approach for avoiding the creation of
practice. Unlike the approach used in traditional

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Educational Sciences: Theory & Practice

teacher-centered education programs (Shor, 1992). Lowrie focused on producing open-ended problems.
Freire (1970) emphasizes that problem posing is a Just as teachers serve as experts when helping students
communication method that easily results in social gain mathematical understanding, they were also
interactions. Using problem posing, teachers can role models during the study’s problem-structuring
obtain appropriate information concerning subjects activities. Teacher–student interactions were also
that students are highly interested in (Freire, 1970). observed, and audio records of the interactions were
captured. The author found that 13 students produced
Problem posing positively affects problem-solving
required open-ended questions at the end of the
capability (Grundmeier, 2003). Therefore, similar
study, emphasizing that they were very excited about
to problem solving, problem posing has also been
problem-posing activities, especially in subjects that
accepted by researchers as the center of mathematics
they liked. It has been established that subjects that
(Silver, 1997). Problem posing requires in-depth
students like the most are the ones for which they
thinking because it is a different way to approach
prefer producing problems, and the consideration of
subjects. Classroom environments and teachers are
this is very important for students during problem
the most significant elements of problem-posing
posing. In addition, it has been found that students’
activities. Students require classroom environments
attitudes toward problem solving are highly correlated
in which they are comfortable, flexible, and
with their teachers’ approaches.
interrogative and in which they are not ashamed
of what they produce during problem-posing
activities; the person who is responsible for creating
Mathematical Creativity and Problem Solving
such environments is a teacher (Moses, Bjork,
and Problem Posing
& Goldenberg, 1993). Students’ critical thinking
abilities are improved by problem posing. Students The questioning of mathematics creativity and
do their best to produce original ideas during defining it by placing concepts in the proper frame
problem-posing activities, thereby enhancing began in 1960 and is still ongoing, although the
their creativity. Then, they begin to pay attention research has not yet identified a universally accepted
to logical relationships and question sentence method for diagnosing mathematics creativity.
formations as they start posing problems. Their Various assessment criteria have been taken into
problem-solving capacities grow more efficient as account concerning this subject. Mathematics
they question whether solutions exist for problems creativity does not refer to forming discoveries from
they create (Cai, 1998; Cankoy & Darbaz, 2010; things that do not exist. Rather, it refers to discovering
English, 1997; Silver, 1997). Abu-Elwan (1999) new connections as a result of formal changes to
conducted a problem-posing study with teacher things that already exist (Ervynck, 1991). Sriraman
candidates to establish that teachers’ support of (2005) defined mathematics creativity as two different
students during problem solving is insufficient; products that emerge from both cognitive processes
teachers must create problems that force students to and result-oriented endeavors. This article focused on
struggle to find solutions, which will improve their how mathematics creativity emerged from cognitive
mathematical thinking capabilities. He concluded processes based on the sampling composition of
that semi-structured problems were more efficient its secondary school students and how the school
for improving problem-posing capabilities. approached creativity in mathematics. The most
distinctive feature of mathematics creativity is that
Lowrie (2002) attempted to compose a conceptual
it entails using original solution methods that differ
frame for five- and six-year-olds that would allow them
from conventional methods (Sternberg & Davidson,
to create their own problems. The study emphasized
2005). However, Romey (1970) states that making
that open-ended questions are relatively much more
original connections among current mathematics
difficult because they have more than one solution.
knowledge, concepts, or approaches can be assumed
Five-week training was given to 25 children from the
to indicate mathematics creativity. Cornish and
1st class in producing open-ended questions, and they
Wines (1980) maintain that adapting well-known
were encouraged in the activity throughout the study.
mathematics knowledge into similar simple events
Although it is known that open-ended questions
can be put into practice because of mathematics
facilitate mathematics understanding more than
creativity. That is, math creativity at school can entail
standard questions, it has been observed that they are
both incorporating traditional mathematics into the
not widely used by teachers and are not incorporated
standards of the developing world and looking for
into textbooks. One way to improve flexible thinking
answers to open-ended math problems and situations
capacity is problem-posing activities. Thus, the study of
(Haylock, 1987).

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Balka (1974) suggested three criteria for math showing more than one way to solve a problem,
creativity: fluency, flexibility, and originality. In and then they assessed their subjects’ geometry
addition, asking the proper questions to find the knowledge and creativity using a geometry
missing information have been the major elements problem. Leikin (2009) stated that finding more
in improving math creativity. Semi-structured than one solution method identifies and establishes
problem posing is one example of this situation. creativity. That is, using multiple solution methods
Carlton (1959) states that the prediction of changes both develops creativity and assists in identifying
based on other changes in the hypotheses of the it. For example, if a student reaches a solution
problems may support mathematics creativity. using a different method from of others, he or she
Structured problem posing, including the What- has a higher level of creativity than do others. In
If-Not strategy developed by Brown and Walter summary, problem posing, mathematics creativity,
(1990), is an example of this situation. Flexible and problem solving have common characteristics.
problem-solving ability has been associated with
math creativity. Asking original questions to
solve problems and presenting solutions from Relational and Instrumental Understanding
various viewpoints have also been referred to as The functionality of knowledge has been questioned
mathematics creativity. since student-based education recently replaced
Sheffield (2008) also reached a similar conclusion traditional education. Education addresses two
that although students who possess mathematics facets of knowledge, instrumental and relational
creativity solve their problems and reach their (Baki, 1998). Instrumental knowledge is the
conclusions using different and original methods, operations that have already been used based on
they also repeatedly study the problems and the certain rules and formulas. The correct application
solutions. Leung (1997) discussed the relationship of algorithms is the main topic rather than seeking
between creativity and problem posing by to answer “why.” In contrast, the meanings of
comparing the characteristics of each concept concepts and the relationships among them are the
and concluded that creativity is in the nature of main topics of conceptual knowledge. Relational
problem posing; that is, creating a problem is a knowledge has been related to symbolizing
creative activity (Leung, 1996). However, Silver math concepts and making meaning out of the
(1994) emphasized that it was not clear that there operations themselves (Soylu & Aydın, 2006).
was a relationship between problem posing and Therefore, it has been suggested that mathematics
creativity. Problem-solving and problem-posing problems should be structured so that they require
activities on research-based math education using both relational and operational knowledge
contributes to student creativity. Understanding (Baki, 1998). Unless the required importance has
the problem, the first stage in problem solving, can been provided for both relational and instrumental
be the beginning of the creative process (Getzels knowledge, there will likely be failures. When
& Csikszentmihalyi, 1962). It can be stated that problems that require instrumental knowledge are
mathematics creativity has a close correlation with solved in the classroom, students do not gain in-
problem solving and problem posing. In addition, depth knowledge concerning abstract mathematics
students’ problem-solving methods that offer more concepts (Bekdemir & Işık, 2007). Related research
than a single solution or their confirmations of their on fractions and both instrumental and relational
results are associated with the flexibility aspect of knowledge have been carefully conducted (Toluk
math creativity (Silver, 1997). & Olkun, 2001). Studies find that the fact that
students do not regularly face fractions and that
In problem posing, the students examine the
they do not conduct relational learning activities
various problems, as well as analyzing them and
that use fractions or make them concrete can cause
writing them up, using their own statements. Silver
fractions to be considered a difficult subject. It has
(1997) maintains that mathematics creativity has
been stated in the literature that students are much
also been correlated with superior intelligence.
more successful with instrumental knowledge
Creativity can be determined by an original compared with relational knowledge. In this
solution to a problem that no one has solved before work, a question that requires both relational and
(Polya, 1945). Levav-Waynberg and Leikin (2009) instrumental knowledge at the same time was used
stated that solving problems using a variety of ways for the study’s problem-posing test. The seventh
can be a marker of creativity and teach-ability. They question on the test was prepared by balancing both
described that geometry has been a proper field for operational and relational knowledge. That is, the

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Educational Sciences: Theory & Practice

correct things had to be carried out correctly and Ben-Peretz, Mendelson, and Kron (2003), studied
the required interpretations had to be realized after 60 teachers of vocational and technical courses and
the operations to solve the actual problem. found that they perceived themselves as a zookeeper, a
maestro, a judge, and a puppeteer. Specifically, teachers
of low-performing students defined themselves as a
Determination of Students’ Thoughts about zookeeper, but teachers of high-performing students
Problem Posing Using Metaphors defined themselves as a maestro.
A metaphor is an essential mechanism of the Frant, Acevedo, and Font (2005) proposed to
mind that lets us know how we think and how investigate the dynamic process of teaching and
we express our thoughts in language (Lakoff & learning graph fiction in high school in Spain.
Johnson, 1980). One of the more effective ways to Researchers sought answers to the following
identify students’ thoughts on problem posing is questions: what kind of metaphors did teachers use
metaphors. Research on metaphors that dates back to explain the graphic representation of functions,
to the work of Aristo (B.C. 386–322) looks at the did the teachers realize the metaphors they used,
use of language and eloquence, and Lakoff and the effect of the metaphors on the students, and the
Johnson (1980) determined that even our mentality role played by metaphors in negotiating meaning.
is formed with metaphors.
Metaphors are widely used in understanding
Metaphors are used as pedagogical, assessment, people’s perceptions in different situations and
and mental tools in education (Saban, Koçneker, different concepts. This study explores students’
& Saban, 2006). They make it easy to conceptualize metaphorical images of problem posing after they
and help to configure knowledge. Most research performed the problem-posing activities.
on using metaphors in mathematics education
presents that metaphors highlight the importance
of education. Metaphors produce a conceptual The Importance of the Study
relationship between a source domain and a target
Researchers such as Cai (1998) and Crespo (2003)
domain because they link different senses (Lakoff &
studied to find correlations between problem
Johnson, 1980). Although conceptually, metaphors
solving and problem posing, and Levav-Waynberg
are related to the person who creates them, teachers
and Leikin (2009) found significant relationships
use them to help increase students’ understanding
between problem solving using different methods
(Lakoff & Nunez, 1997). That is, in discussions of
and creativity. Nonetheless, no researcher has
abstract concepts, the use of metaphors provides
investigated the link between problem solving
the coherence of meaning.
by multiple methods and problem posing.
Metaphors are experiences that are acquired from Furthermore, students’ views of problem posing
our daily lives, and they are conceptual. They are were investigated through metaphor analysis.
indispensable for comprehending abstract notions. Establishing how students (gifted and non-gifted)
Because of their conceptuality, metaphors are shaped use the particular method of metaphors to pose
according to different cultures (Lakoff & Johnson, problems will contribute to the literature.
2005). Picker and Berry (2000) asked seventh- and
Problem posing has been studied by many
eighth-graders five different elementary schools in
researchers in mathematics education. However,
five different countries to draw their mathematics
this study was the first in the literature to use a
teachers. The analysis of the metaphors revealed
problem-posing activity with multiple choice
that students had drawn threatening, violent,
questions; when the test questions were being
despotic, and rigorous figures. In the 1996 study
developed, misleading options were inserted, thus
by Inbar, 409 primary students and 254 educators
requiring students to confront both problem posing
participated offered metaphors for the concepts of
and problem solving. In addition, the students’
student, teacher, and school principal.
teacher no longer has to lose time attempting to
Students were perceived as vegetation by many of determine how to evaluate posed problems.
the educators, and the educators were perceived as
a super power by many of the students. The school
principal was conceptualized as an authority figure Research Questions:
by the students and educators, and they conceived 1. Is there a significant relationship between
of school as being framed by the world. problem solving ability using multiple methods
and problem posing ability?

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 Arıkan, Ünal / Investigation of Problem-Solving and Problem-Posing Abilities of Seventh-Grade Students

2. Is there a significant dependence between academic gains without classifying them by the
multiple problem solving and giftedness? fields in which they are more talented. The classes
are referred to in the literature as “complete special
3. What are gifted and non-gifted students’
classes.” The “complete special class” environmental
metaphorical images of problem posing? What
factor has been recognized as important for
differences, if any, exist between gifted and non-
talented students to make academically defined
gifted students’ metaphorical images of problem
gains (Rogers, 2002). Talented individuals show
posing?
extraordinary performance in at least one field and
put their signatures on creative ideas. They mature
Research Design earlier than their peers and continue to mature
and develop skills well past the time when others’
The main goal of this study was to investigate skills have peaked. For example, whereas normal
the effect of multiple problem-solving skills on individuals might progress during a defined time
the problem-posing abilities of gifted and non- period, more talented individuals continue to
gifted students. Another goal was to explore these progress until much later ages (Winner, 1996).
students’ metaphorical images of problem posing. It has been defined in the literature that talented
The research model of this work was a survey, a students can easily understand concepts, show
descriptive model that aims at describing situations flexible thinking, are open to exploring new things,
without interfering with or changing the situations. examine the details, and possess high levels of
Non-experimental research is conducted in natural ethical sense (Reynolds & Birch, 1988).
settings, with numerous variables that operate
simultaneously. This study was designed to seek
the answers to the research questions by employing Data Collection Tools
both quantitative and qualitative techniques.
The problem-solving task consisted of five fraction
problems to be solved in multiple ways. The test was
Participants constructed as 10 questions for the pilot application,
and it was presented to experts for review. The
Eighty-five non-gifted public school students and problems that were produced by the researcher were
20 gifted private school students, all in the seventh designed to be solved in three ways, arithmetic,
grade, participated in the study. Seventh grade visualization, and algebraic. The students were
was chosen. Because these students could solve asked to choose five problems and solve them
fraction problems not only arithmetically but also using more than one solution in the pilot. Because
algebraically; the participants had learned algebraic the students in the pilot study could solve the
solutions in the sixth grade following the secondary problems in items 1, 3, 5, 9, and 10 in more than one
school mathematics curriculum. way, these problems were selected and used in the
The gifted students, who were enrolled in a full master work. The reliability of the test using these
special class, were drawn from two private schools. selected items was calculated as .857 in the pilot
Criterion sampling was used, and all of the gifted study. Divided test solutions were implemented
students who participated in this work had obtained for internal consistency, and the Cronbach’s alpha,
scores of 135 or above on the Wecshler Intelligence Spearman-Brown, and Guttman coefficients were
Scale for Children (WISC-R). The WISC-R is one calculated (.714, .833, and .809, respectively). The
of the most common scales for assessing giftedness problem-solving test was determined to be reliable,
(Savaşır & Şahin, 1995). The non-gifted sample and it showed internal consistency.
was drawn from one public school in Istanbul. The problem-posing task consisted of twelve
Convenience sampling was used for selecting the multiple-choice items. Below is the problem-
non-gifted participants. The students were easy posing question, which came from page 43 of the
to recruit, and the researchers did not consider third-grade mathematics textbook (Erbaş, 2014)
selecting participants who were representative of published by the Turkish Ministry of National
the entire population. Education for school year 2013–2014, which served
The full special class is very important gifted as the infrastructure for this study.
students’ learning (Rogers, 2002). Considering From page 43 of the 3rd-grade mathematics
the environmental factors (class selection, school textbook published by the Turkish Ministry of
selection, etc.), special classes have been dedicated National Education, 2014
to exceptional students to contribute to their

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Educational Sciences: Theory & Practice

Table 1
The Study’s Problem-Posing Question
In Turkish In English
Which one of the below problems can be matched with the
operation of
213 + 167 = 380?
A) Osman picked up 213 pieces of walnut. Recep picked up
167 more pieces of nuts more than Osman. What is the total
amount of the nuts that both Osman and Recep picked up?
B) On Saturday, 213 and on Sunday 167 bottles of water
were sold in a market. What is the total number of bottles of
water that were sold at this market on these two days?
C) Erdem has 213 Turkish lira. His brother has 167 lira less
than that. What is the total amount of money that both
Erdem and his brother have?

The aim of this study was to investigate one factor possible solutions. The semi-structured and
that affects problem-posing capability, problem structured problem-posing situations were used
solving using multiple methods. The participants in both applications and were exactly the same.
were 105 seventh-grade students, twenty of whom However, the problems with only the multiple
were gifted. Two achievement tests, one on problem selections were prepared by the researchers and
posing and one on problem solving, were used as a presented to the students as ready problems with
data collection tool. related selection options.
The experts were two Turkish teachers and four Improper problems, impossible cases, unnecessary
math teachers who were asked to validate the tests. or excessive knowledge and other problems were
The problem-posing test was multiple-choice. To the main misleading options on the problem-
provide internal consistency, split test analysis was posing test. Furthermore, 5 of the problem-posing
used on both tests. tasks were structured, and the remainders were
semi-structured situations.
We prepared the problem-posing test based on a
strategy that was developed by Stoyanova and Ellerton The achievement tests were reformulated during
(1996). The test items were designed using semi- the design phase by two teachers who were experts
structured and structured problem-posing situations. in their fields. The opinions of the above-referenced
experts were considered during the pilot study, and
The problem-posing test was developed in two
the group confirmed that the problems on both
stages. First, we investigated whether the problem
tests could be correctly understood by the students
posing was realized with multiple questions and
and that they met the students’ cognitive levels. The
also whether there was a difference between the
pilot study was conducted 100 students, 10% of
classical problem-posing operations and other
whom were gifted.
methods. The situations were presented to the two
teachers who were experts in their fields at the state The reliability of the achievement tests was checked
university in addition to two other math teachers with the pilot study, and the value for the problem-
and one Turkish teacher at the school where the posing test was .855 (p < .05). Divided test solutions
study was conducted. The experts presented their were implemented for internal consistency, and the
predictions regarding whether administering either Cronbach’s alpha, Spearman-Brown, and Guttman
of the tests would pose any problems for seventh- coefficients were calculated, at values of .706, .846,
grade students. This study also used descriptive, and .844, respectively. The problem-solving test was
non-empirical research methods to supplement found to be reliable and showed internal consistency.
the quantitative approaches. The semi-structured
Direct observation was also used in this as one of the
and structured problem-posing situations that were
methods to support the quantitative findings. The
created by Stoyanova and Ellerton (1996) were used
metaphors created by the students were classified
here, in the following formats:
through content analysis. The most salient findings
a) Constructing fractions in the correct format from the study were that: the gifted students could
for the given operation not produce entirely new ways of problem solving;
the students who did finding multiple solutions
b) Changing the data on a given problem
had higher scores on the problem-posing test; and
The 68 students from the state school first the metaphoric thoughts in the problem-posing
faced a classically structured problem-posing activities have much more positive effects on the
situation and then a situation with multiple normal students.

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Data Analysis analysis uses inductive reasoning, by which themes

and categories emerge from the data through the
To date, many problem-posing studies have been
researcher’s careful examination and constant
conducted according to classical practices. In
comparison. To organize the data in this study, the
other words, researchers have evaluated problems
students’ metaphors were listed and grouped into
that were posed by students according to certain
categories, and the data were analyzed qualitatively
evaluation criteria. Leikin and Lev (2013)
in three phases. The metaphors were independently
evaluated posed problems in terms of correctness,
a) coded by concept, b) classified by topic, source,
creativity (fluency, flexibility, and originality), and
and connection between topic and source, and c)
connectedness. Silver and Cai (1996) evaluated
examined for common characteristics. Subsequently,
posed problems according to correctness and
the researchers compared their lists of metaphoric
semantic or linguistic difficulty. In our study, we
images and found the least common denominator.
aimed to identify the link between problem solving
using multiple methods and problem posing.
Therefore, we prepared the problem-posing task as a
Results
multiple-choice test. We preferred to use distracters
in options, as did Singer and Voica (2012). Hence, The findings are discussed in the order of the
we did not establish any evaluation criterion; we research questions. The first question aimed to
only assessed the correctness of problems posed explore the relationship between multiple problem
by students. One point for the correct answer and solving ability and problem posing ability.
two points for the alternative solution were given One of the questions that remains in the literature is
in problem-solving tests during the analysis of the whether students who are successful in finding and
metaphor data. One point was given for correct producing alternative solutions are also successful
answers in the problem-posing test. A chi-square at problem posing, that is, for this study, whether
test with Yates’s correction for continuity was there was any correlation between problem-solving
performed to determine dependency between and problem-posing capabilities. The Pearson
giftedness and problem solving using multiple ways. correlation coefficient was r = .760 when the
After the achievement tests were administered, the students’ responses were analyzed in SPSS 18.0.
results were analyzed using SPSS 18.0 software. The Thus, there was a strong correlation (p = .00 < .01).
problem-solving test items were graded, presented
in Table 2: Table 5
Correlations between Problem Solving and Problem Posing
Correlations
Table 2
Problem- Solving Test Grading Problem Pearson correlation Sig.
1 .760***
posing (2-tailed)
Points Description
Problem Pearson correlation Sig.
O No answer or incorrect answer .760*** 1
solving (2-tailed)
1 One way (single) solution
2 Multiple (alternative) ways of solution Note. N = 105, correlation is significant at the .001 level
(2-tailed).

The frequency distribution for solving problems

One hundred and five students, 20 of whom were
using more than one was assessed following the
gifted, participated in the work. Eighteen students
problem-solving test. The relationship between
presented at least one alternative way of solving
the students’ problem-solving and problem-posing
the 5 fraction problems; surprisingly, not all 18
test responses were checked with the Spearman
were gifted, only 13 of them. The distribution of
correlation coefficient. Because it is not a parametric
the students who solved at least one question on
measure, it was used as a special state of the Pearson
the problem-solving test using multiple methods is
correlation coefficient.
shown in Table 3.
Yates’s correction for continuity is mostly used
when at least one cell in a table has an expected Table 3
count smaller than 5 (Yates, 1934). This study used Distribution of the Answers Given on the Problem-Solving Test
the chi-square test to identify the dependence Multiple Ways Only One Way
between multiple problem solving and giftedness. Gifted Students 13 7
Non-Gifted Students 5 80
Content analysis was used on the metaphors that
were generated by the students. Qualitative content

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Educational Sciences: Theory & Practice

Table 4
Different Way of Asking the Same Type of Question
Problem posing with interpretation Problem posing by modifying the problem
“The sum of the ages of both a. The problem is correct. “The sum of the ages of both Ahmet and a.55/ 3 b.56/ 2
Ali and his father is 54. If Ali’s b. The problem has missing in- his mother is 45. If Ahmet’s age is 2/7 of 7 5
age is 4/5 of his father’s age, formation. his mother’s age, how old is Ahmet?” 1 d.52/ 6
how old is Ali?” c. The problem has unneces- How could we re-pose this problem by c.52/
4 7
Which one of the selections sary information. changing the fraction and the total age?
in the right column is correct? d. The problem is impossible. e.50/ 2
5

Whereas 65% of the gifted students gave the correct Researcher: But you are a gifted student. How could you
results on the problem-solving test, only 5.88% of hate mathematics? You have a high WISC-R score.
the non-gifted students did so. Yates’s correction
Gifted Student: But we were not asked mathematics
for continuity value was .78. Because the degree of
on the WISC-R test. It is related to reasoning.
freedom in Table 3 is 1, χ20,01;1 = 6.63490 < 35.78,
and therefore, giftedness and problem solving by Metaphors are clues that are related to a person’s
multiple ways were dependent variables. We also ideas about a concept (Levine, 2005). We wanted
investigated the coefficient of contingency where n = to know students’ thoughts about problem posing,
sum of observed values. That coefficient was 00.707. we were surprised to find that the gifted students
generated negative metaphors related to problem
Five structured and 7 semi-structured situations
posing. We asked them to provide their thoughts
were presented on the problem-posing test, and the
and experiences concerning the activities because
students were asked to comment on the problems
their metaphors reflected their daily life experiences
that comprised the four structured problem-posing
(Lakoff & Johnson, 2005).
questions. In addition, the students were asked one
question in the manner presented below, which Forty percent of the gifted students found the
includes both interpretation and modification. problem posing unnecessary when the produced
In fact, this situation can be considered a way to metaphors were examined carefully, but only
increase the test’s reliability. 3.8% of the non-gifted students did so. Moreover,
the gifted students who could not produce any
alternative solutions were among those who found
Notes from Direct Observation: Non-gifted the problem posing unnecessary.
students were observed to greatly enjoy the
It was observed that 30.76% of the gifted students
problem-posing task. However, they reported that
who solved the problems in more than one way
they would have liked it even more if the test had
also considered the problem solving to be an
not included. In addition, some students did not
unnecessary activity. The same percentage also
solve the problems through visualization. When
used metaphoric statements that revealed that the
they were asked, they said that visualization was a
focusing and the infrastructure which requires
fourth-year subject.
the capability and the experience have been
Three gifted students were not willing to solve significantly important. Another important point
problems and said that they hated mathematics. in the table is that no gifted students made mention
This researcher interviewed one of them: of a lack of experience, which indicated that the
students had previously posed problems.

Table 6
Categories of Generated Metaphors by Students

Categories of Gifted Students Non-Gifted Students

Metaphors
Solving a problem in Not solving a problem Solving a problem in Not solving a problem
multiple ways in multiple ways multiple ways in multiple ways
1 Inexperience - 15
2 Needlessness 4 4 - 4
3 Complexness 2 - 3 24
4 Time-consuming 2 - 2 11
5 Demanding 4 - - 18
6 Funny 1 - - 18
7 Necessity 3 - - 10

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 Arıkan, Ünal / Investigation of Problem-Solving and Problem-Posing Abilities of Seventh-Grade Students

Non-gifted students who solved the math alternative solution to any question. Meanwhile,
mathematical problems in more than one way found students must be given the opportunity to solve
the problem posing to be difficult and time consuming problems in alternative ways and to produce
and to require patience. It was noted in examining the problems in their own languages.
table that most of the normal students who could not
Arıkan and Ünal (2012) determined that eleventh-
produce any alternative solutions found that using
grade students were satisfied with only one solution
metaphors to pose problems was complicated and
rather than seeking alternatives; that is, both high
difficult. Table 6 summarizes the metaphors generated
school and secondary school students preferred to
by both non gifted and gifted students whom were
just solve the problems, not to pose new ones. It was
able to solve a problem in multiple ways and not
observed here that the curiosity and the eagerness
able to solve the problems in multiple ways. Student
of the gifted students could be inferred from their
metaphors were fell into seven categories. The purpose
scores on both the problem-solving and problem-
was to explore students’ perceptions of problem
posing tests. Their tendency to solve problems in
posing through the lenses of metaphors. For instance,
multiple ways was more pronounced than the same
while one gifted students found the problem posing
tendency in the non-gifted students. Yates’s chi
activity as enjoyable, 18 non gifted students found
square result determined giftedness and problem
the problem posing as enjoyable as shown row 6 in
solving in multiple ways to be dependent variables.
Table under the Funny category. It was interesting that
Hence, giftedness can be examined using problem
gifted students whom were able to solve the problem
solving by multiple ways.
in multiple ways, their perception of problem posing
varied from necessary activity to time consuming The same type of problem-posing question was
activity. Furthermore, non gifted student’ perceptions used in both interpreting and implementing the
whom were not able to solve the problem in multiple data modifications. Thus, it was possible to present
ways were mixed, but they found the problem posing the problem-posing situations with misleading
activity useful and necessary. choices. This study examined whether there was
a correlation between the capabilities of both
That is, the normal students, whether they identified
problem solving and problem posing and found
alternative solutions or not assess problem posing
a robust correlation. It can be inferred that the
as complicated.
students who solved problems in multiple ways
will also be more successful in posing problems,
which supports the findings by Arıkan and Ünal
Discussion and Conclusion
(2014) and Cai (1998). In fact, there has been no
Multiple problem solving and being gifted were significant evidence in the literature that there is
observed as the dependent variables in this always a correlation between problem solving and
research, which supported the study by Levav- problem posing. On the contrary, another study by
Waynberg and Leikin (2009). In addition, a strong Crespo (2003) did not accept that there was such
correlation between multiple problem solving and a correlation, and thus, whether this relationship
problem posing capability was revealed. It can be exists is still under debate.
concluded that it would be useful to encourage
Most of the non-gifted students did not like having
students to solve math problems in different ways.
to work with fractions, and found it difficult to pose
Problem-posing activities should be described to
problems because of their lack of experience. It can
teachers during their in-service trainings and their
be suggested that teachers should use the required
importance should be emphasized accordingly.
materials on fractions as often as possible because
Separately, it could be useful to establish whether
so many students were not happy about working
teachers in Turkey have used in-class problem-
with them. It was found in the study that some of
posing activities.
the students were confused by both compound
It is stated in the curriculum, which has been in and simple fractions and could not tell the exact
development since 2006, that problem-posing difference between them; that is, they saw no
activities are as important as problem-solving difference between ¼ and 4/1. In addition, it was
activities. This situation shows the importance of observed that the students assumed that the fraction
presenting problem posing to teachers during in- solutions they had learned in the 4th grade did not
service trainings, although time could be an issue relate to each other, and they perceived that the
when the teachers have a syllabus they must follow. lessons were only for the 4th grade. For example,
Most of the students, 87%, could not produce any the students could not remember to use modeling,

1413
Educational Sciences: Theory & Practice

i.e., using boxes or drawing figures or shapes to (1998) depicted these students as solving math
arrive at a solution; even those who remembered problems quickly and using different strategies to
to use the technique were prejudiced against it and solve the same problem. Given that only 13 out
felt that it mainly related to the 4th grade lessons. of 20 students in this study could solve problems
It can be concluded from this work that there were in multiple ways, the following question is raised:
still significant defects in the students’ practice “Does giftedness necessarily mean giftedness at
even with the adoption of a structuring rather than mathematics?” We might benefit from specific
memorizing approach even if that is required in the instruments to identify giftedness in mathematics.
syllabus. It appears that the students in this study
Based on the results of the content analysis of the
had merely memorized the subjects when they first
metaphors, it could be the case that the gifted
learned them and no longer considered them after
students found it dull and uninspiring to choose
they completed that grade because they thought they
from limited response options and instead preferred
would not need them anymore. Therefore, it appears
to construct and pose problems. The majority of the
that it would be very useful to teach fractions using
non-gifted students, however, found constructing
real-life connections and building relationships
problems difficult and complicated, and they used
in the materials because fractions comprise the
metaphors to reflect their inexperience.
infrastructure of numerous math subjects. Teachers
can assess the students’ common mistakes and
conceptual misunderstandings, taking advantage of Recommendations for Future Research
the current technology and creating environments in
students have the opportunity to assess themselves The complexity of the relationship between problem
based on their work. One of these assessment posing and problem solving was not completely
methods is problem posing, which gives teachers addressed in this study; much more remains to be
information on students’ strong and weak points learned. For example, studies are needed to examine
after they check the questions the students pose. problem posing and problem solving in different
branches of mathematics such as geometry and
Levav-Waynberg and Leikin (2009) said that probability. From the teaching perspective, the role
multiple problem solving is used to assess gifted of classroom activities in building problem-posing
students. Holton and Gaffney (1994) emphasized skills and that of instructions in the process should
that mathematically gifted students take pleasure be investigated.
in numbers and mathematical subjects, and Villani

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